Browsing by Author "Schilling, Anne"

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    Crystal energy functions via the charge in types A and C
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Lenart, Cristian; Schilling, Anne
    The Ram-Yip formula for Macdonald polynomials (at t=0) provides a statistic which we call charge. In types A and C it can be defined on tensor products of Kashiwara-Nakashima single column crystals. In this paper we prove that the charge is equal to the (negative of the) energy function on affine crystals. The algorithm for computing charge is much simpler and can be more efficiently computed than the recursive definition of energy in terms of the combinatorial R-matrix.
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    An uniform model for Kirillov-Reshetikhin crystals I: Lifting the parabolic quantum Bruhat graph
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2012) Lenart, Cristian; Naito, Satoshi; Sagaki, Daisuke; Schilling, Anne; Shimozono, Mark
    We consider two lifts of the parabolic quantum Bruhat graph, one into the Bruhat order in the affine Weyl group and the other into a level-zero weight poset first considered by Littelmann. The lift into the affine Weyl group gives rise to Diamond Lemmas for the parabolic quantum Bruhat graph. Littlemann's poset is defined on Lakshmibai-Seshadri paths for arbitrary (not necessarily dominant) weights. Here we consider this poset for level-zero weights and determine its local structure (such as cover relations) in terms of the parabolic quantum Bruhat graph. Littelmann had not determined the local structure of this poset. In addition, we show a generalization of results by Deodhar, coined the tilted-Bruhat theorem, which involves the compatibility of the quantum Bruhat graph with the cosets for every parabolic subgroup of the Weyl group. We will use the results in this paper in a second paper to establish the equality between the Macdonald polynomials Pλ(q,t) specialized at t=0 and Xλ(q) which is the graded character of a simple Lie algebra coming from tensor products of Kirillov-Reshetikhin (KR) modules.